Quantum Non-Demolition in Optics Novice Pedagogical Review by Nergis Grangier et al, Nature 398 (1998) Chiao et al, quant-ph 9501016 (1995) Buchler, Ph.D. thesis, ANU (2001)

Quantum mechanics basics Non-commuting operators Obey Heisenberg uncertainty principle Operator terminology standard deviation, dispersion, uncertainty variance expectation value

Measurement Back Action Heisenberg Uncertainty Principle Measure A very precisely  DA arbitrarily small  DB very large Large DB directly imposes no restriction on precision of A unless the large fluctuations in B couple back to A Precision measurement introduces a perturbation or ‘back action’ How to evade this back action ‘noise’?

QND to the rescue Devise a measurement where back action noise is kept entirely within unwanted observables and without being coupled back to quantity of interest Types of QND Mechanical oscillators (GW bars, NEMs) Quantum Optics

Is it QND? Pedagogical example Stern-Gerlach (SG) apparatus SG filter A good quantum state preparation (QSP) Once –/2 is filtered out and there are no perturbations, only ever get +/2 +/2 -/2 e- +/2 -/2 e- SG filter

Is it QND? But what if +/2 is perturbed between two SG filters? If the perturbation flips +/2 to –/2, nothing passes through the next filter and the +h/2 state is “demolished” Filter: selects a quantum state with given eigenvalue QND device: measures the quantum state and keeps the particle going regardless of the outcome of the measurement  SG filter  QND device since +/2 state prepared by demolishing –/2

Good QND observables? Spin Generally, constants of motion Without external perturbation spin doesn’t evolve between successive measurements Generally, constants of motion Position, e.g., is not ideal since Dp  Dx(t)

QND schemes need a readout Even if a good QND variable (e.g. spin) exists still need to ‘know’ (measure) its eigenstate Measuring eigenstate destroys it READOUT to the rescue Couple the spin of test particle to the spin of a meter or probe particle The coupling is such that it does not affect signal particle’s eigenstate Measure the spin of the meter particle (so what if meter particle’s eigenstate is destroyed)

EPR paradox  measurement of ym gives ys Entangled States EPR paradox  measurement of ym gives ys +/2 -/2 meter particles signal interaction regions Entangled state y = ays + bym t0 t1 Entangled state  simultaneous eigenstate of deltaX and SigmaP. Non-factorizable superposition of product states. EPR  measure X (or P) of particle 1, predict with certainty X (or P) of particle 2 even if separation is greater than light cone (space-like intervals) Aspect expt two blue photons emitted from atomic transition with 0 net angular momentum  orthogonal polarzations. Coincidence count rate as function of relative angle between polarizers in the two beams is measure of the correlation between two well separated photons. Once meter particle is detected, all other non-commuting observables of the signal particles are randomized but that’s okay since they are not back coupled onto original QND variable

QND in Optics Squeezed light paved way for manipulation of quantum noise of light QND used to control of the light quantum noise Standard technique Signal and meter beams coupled via a non-linear optical medium Usual observables Quadrature amplitudes of (quantized) EM field Photon number and phase signal beam meter beam non-linear medium

Quadrature Fields Add a steady carrier field E0 to fluctuating electric field with quadratures E1 and E2  ET(t) = E0 + E1(t) cos(w0t) + E2(t) sin(w0t)  E0 (1 + E1(t)/E0) cos[w0 (1 + E2(t)/E0) t] E1(t) gives amplitude modulation E2(t) gives phase modulation   fluctuating part is small (linear approximation)

Quantization of quadrature fields E-field operators using two-photon formalism (W = light frequency and j = 1,2) Ej(t) = (aj e-iWt + aj+ eiWt) dW/(2p) Commutation relations [E1(t), E1(t’)] = 0 [E2(t), E2(t’)] = 0 [E1(t), E2(t’)] = i(t – t’) E1(t), E2(t)  X(t), Y(t)  position, momentum operators of QHO

Standard Quantum Limit Precision scale set by shot noise limit  Dn = n DX Df = (1/2 n) DY Heisenberg  Dn Df  1/2  DX DY  1 Standard Quantum Limit DX DY = 1

Is it QND?  QSP criterion  Measurement efficiency criterion After measurement is performed, system should be left in a well-defined eigenstate Quantified using conditional uncertainty in DXs after measurement of m, DXs|m For laser beam, limit set by shot noise  DXs|m < 1  Measurement efficiency criterion Noise of measurement device Quantified by ‘input’ refered noise’ of meter, DXm  Non-demolition criterion How much is measured observable disturbed by the measurement? Quantified by signal noise, DXm

QND quantified , ,   efficient measurement minimizes DXs and DXm  DXs DXm < 1 T is the transfer function of signal-to-quantum-noise for s and m, respectively Ts = 1/(1+DXs2) Tm = 1/(1+DXm2) Then DXs DXm < 1  Ts + Tm > 1 General conditions for a QND measurement Ts + Tm > 1 (DXs|m ) 2 < 1

QND in Optics Need non-linearity to couple signal and meter beams Second-order non-linearities Parametric amplification  amplifcation of signal beam at w using a pump beam at 2w Produced with special crystal, e.g. KTP Third-order non-linearities Kerr effect  intensity-dependent refractive index Cross-Kerr effect  n for one beam is modified by intensity of a second beam Produced in fibers, resonant 3-level (cold, trapped) atoms

Second-order non-linearities Parametric processes useful in QND allows meter beam to carry amplified copy of signal beam How? Noiseless amplification Amplify one quadrature of signal while preserving SNQR (i.e. Ts  1) Relative phase between signal and pump beam (Conventional optical amplifiers (lasers) have Ts = 0.5 for large gains  3 dB loss in SQNR) Noiseless amplification Deamplification  shot noise limited light becomes squeezed

Third-order non-linearities Cross-phase modulation  index of beam 1 modulated by intensity of beam 2 n(s) = n0(s) + n2(s) I(m) n(m) = n0(m) + n2(m) I(s) Gain  g2 = (Fs Fm) where Fs,m = ks,ml n2(s,m)I Ts = 1  input beam is shot noise limited Tm = g2/(1 + g2) (DXs|m)2 = 1/(1 + g2)  perfect QND as g   Sout Mout

QND(3) Input-output transformations for quantum fluctuations of the beams XOUT(s) = XIN(s) YOUT(s) = YIN(s) - g XIN(m) XOUT(m) = XIN(m) YOUT(m) = YIN(m) - g XIN(s) Intensity noise of meter beam XIN(m) is fed back onto phase noise of signal beam YOUT(s) Measurement back action  variance of signal phase noise (DXs|m)2 = 1 + g2  Heisenberg or worse as g > 0 Conditional variance of signal intensity noise  (DXs|m)2 = 1/(1 + g2)  Heisenberg or better

Present state of QND (DXs|m)2